• The Book of Why: The New Science of Cause and Effect (Pearl and Mackenzie, 2018)
• link to article "How a Pioneer of Machine Learning Became One of Its Sharpest Critics - The Atlantic"

• Methods for estimating causal effects: how to answer the question of "What is the effect of A on B?"
• Randomized designs
• Alternative designs when randomization is infeasible: matching methods, propensity scores, regression discontinuity, and instrumental variables. When are they most feasible?
• Broad spectrum of applications: health sciences, genetic studies, mental health, econometrics, public policy, education, social sciences … (That's why causal inference is very exciting)
• But…we need to define causal questions first.

• For a woman older >50 yrs, should she be getting regular screening for breast cancer?
• Do citizens of Los Angeles die because of air polution?
• What is the effect of heavy adolescent marijuana use on adult outcomes (e.g., earnings at age 40)?
• How much mortality and other burden was due to tobacco industry's misconduct?
• Does the Head Start program improve educational and health outcomes for children?
• Does a "healthy marriage" intervention improve relationship quality?
• How does the type of school affect a child's achievements later in life?
• …

• Are parents more conservative than their children because they are older?
• Is there an effect of gender in this regression?

(Hint: "effect of cause" or "cause of effect")

A causal question is a problem with a manipulable intervention.

• Important (and hot) topic right now
• Comparative effectiveness
• Debates regarding study design: "efficacy" versus "effectiveness"; observational versus randomized experiments
• Analytic challenges on modern study designs.
  • Community-level interventions (highway billboards, vaccination, new program…); matched-pair cluster-randomized trials (Wu et al., 2014, Biometrics; It's me)
  • Facebook A/B testing which ordering of ads/friends' posts makes you click
  • …

• Be able to formalize causal effect discussions
• Understand key elements of causal inference
• Resolve the Lord's paradox

• What is the effect of some "treatment" \(T\) on an outcome \(Y\)?
  • Effect of a cause rather than cause of an effect
  • \(T\) must be a particular "intervention": something we can imagine giving or withholding
  • e.g. smoking a pack a day on lung cancer, Good Behavior Game on children's behavior and academic achievement

• Units, at a particular place and time
• Treatments/interventions to compare (e.g., \(T=0\) for standard, \(T=1\) for new treatment)

• Potential outcomes, e.g., \(Y_i(1), Y_i(0)\) are the outcomes that would be observed on the same subjects if assigned new, or alternatively, if assigned standard treatment

• Causal Effect (definition): comparisons of potential outcomes for the same subject or same groups of subjects.

Help us be very clear about the effects we are estimating. It helps create a data table with observed and unobserved potential outcomes.

• The entity to which we apply or withhold the treatment
• e.g., individuals, schools, communities
• At a particular point in time
  • Me today and me tomorrow are two different units
• Example: adolescents, elderly people

• The "intervention" that we could apply or withhold
  • Not "being male" or "being black"
  • Think of specific intervention that could happen
  • Example: Body mass index (BMI); heavy drug use during adolescence; trained nurses in a clinic to help manage the care for elderly people
• Defined in reference to some control condition of interest (!)
  • Defining control could be more difficult than the treatment
  • No treatment? Existing standard treatment?
  • Example: no or light drug use?

• The potential outcomes that could be observed for each unit
  • \(Y(T=1)=Y(1)\): the outcome that could be observed if a unit gets the treatment
  • \(Y(T=0)=Y(0)\): the outcome that could be observed if a unit gets the control
• For example, your headache pain in two hours if you take an aspirin; your headache pain in two hours if not taking the aspirin
• Example: earnings if are heavy drug user (\(Y_i(1)\)); earnings if not (\(Y_i(0)\))
• Causal effects are comparisons of these potential outcomes
• No causal inference when exsitence of both \(Y_i(1)\) and \(Y_i(0)\) makes no sense.

• Estimate causal effect:
  • \(ATE = 1/N\sum_{i=1}^N Y_i(1)-Y_i(0)\)
  • That is: \(\left(\sum_{i: T_i=1}Y_i(1) - {\color{red}\sum_{i:T_i=1}Y_i(0)}+{\color{red}\sum_{i: T_i=0}Y_i(1)} - \sum_{i:T_i=0}Y_i(0)\right)/N\)
• Problem: cannot observe both \(Y_i(1)\) and \(Y_i(0)\)
• Your goal is to estimate
  • \({\color{red}\sum_{i:T_i=1}Y_i(0)}\) and
  • \({\color{red}\sum_{i: T_i=0}Y_i(1)}\)
• What does randomization ensure?

• Replication
• The Stable Unit Treatment Value Assumption (SUTVA)
• The assignment mechanism

• Need to have multiple units, some getting treatment and some getting control
• The number of potential outcomes grows

• No interference between units: treatment assignment of one unit does not affect potential outcomes of another unit.
  • Agricultural experiments (guard rows)
  • Drug use of one individual does not affect earnings of other individuals
• Only one version of each treatment
  • Lumping all "heavy" drug use together; estimating effect of any "heavy drug use"

• Process that determines which treatment unit receives
• Randomized experiments: Known (nice!) assignment mechanism
• Observational studies: have to posit an assignment mechanism (determine/model why some individuals become heavy drug users)
  • Propensity score models the assignment mechanism

• High-level: Assignment mechanism is the rule (possibly probabilistic) by which subjects get their actual treatments \(\{T_i\}\). The assigned treatments unmask the potential outcomes \(Y_i(T_i)\) (denoted by \(Y_i^{obs}\)), but mask the rest of potential outcomes, denoted by \(Y_i^{miss}\)
• Central to causal inference (look for it when reading literature; many not mentioning this)
• An extreme example: a doctor who always gives her patients the best treatment (no randomness given patient information and the doctor)
  • but over a population of doctors, the assignments can be summarized probabilistically.

Consider the framework:

• Units:
• Covariates:
• Potential outcomes:
• Treatment:
• Control:

Consider the framework:

• Units: students
• Covariates: Sex, September weight
• Potential outcomes: June weight under treatment and control
• Treatment: University diet
• Control: ???

Pearl (2016) Lord's Paradox Revisited - (Oh Lord! Kumbaya!)

• Understand causal problems as potential interventions and how they are different from association questions
• Understand the framework to discuss/mathematicize causal inference (potential outcomes, assignment mechanism); graphical approaches (Pearl) not discussed here but hugely important for their intuitive appeals and robustness to probability specifications ("The Book of Why")
• Understand designs (ways to efficiently collect useful data) and methods for analyzing these data to answer causal questions
• Understand complications in causal studies, including missing data, noncompliance and many hidden biases.
• Use these as powerful tools to critically review research with causal claims. And improve science!

• Causal inference is counterfactual
• Causal effect is a function of \((Y(1), Y(0))\)
• Causal inference requires estimation of unobserved reponses - it makes sense when the estimation does
• Causal inference requires assignment mechanism
• Assignment mechanism known in randomized studies; must be assumed or modeled in observational studies (propensity scores)
• Causal inference makes assumptions, e.g., non-interference, etc.

• Thank you!
• zhenkewu.com
• zhenkewu@umich.edu